ONE NEAT PROOF! Deriving the EULER DEFINITION of the Gamma Function!

[@stimpyfeelinit]

I HOPE YOU AGREE WITH ME *vague germanic threat*

[@ryanmuller9497]

That moment when Papa Flammy defines n as being strictly less than z and then takes the limit as n approaches infinity while keeping z finite... (I assume that there's a second case to the derivation in which n is greater than z that ends up with the same result at some point down the track, but still a bit slack :P )

[@מיכאלקונטרוביץ]

but generally z is a complex number,so z>n is meaningless,untill we are speaking of the modulus

[@sergiokorochinsky49]

This reminded me a class in Materials Science when the teacher approximated sin(theta) by theta for very small angles, and then proceeded to integrate between 0 and 2pi. :-)

[@epicmorphism2240]

@arabicboi Yes you‘re right but than it also wouldn’t make sense to define n < z. Please correct me if I‘m wrong but I think this proof is not valid.

[@rish5827]

@@epicmorphism2240 I don't think it's meant to be an entirely rigorous proof, just sort of showing the general ideas

[@epicmorphism2240]

Rish There is a difference between rigorous and wrong…

[@bluebears6627]

"If you have one apple then you still have one apple" -one random flammy boi

[@Silver_G]

10:01 About that n = [ 2/1 • 3/2 • 4/3 •…• n/(n-1) • (n+1)/n], shouldn't it be left with n + 1 after those cancellation?

[@OunegNebty]

Yes I Had the same question papa

[@thomaskim5394]

That needs to be more clear. Just saying that it does not matter in limit would not be enough.

[@Downloader77]

Well spotted @Silver-G

[@reetanshukumar1865]

it should be n/n-1 in the end

[@a.e.6994]

It's a very good question. But see in the video by min 10:24. He changed (n+1)/n into 1+(1/n). In this version let n run to infinity. So you have n=n. The idea of reetanshu is also remarkable. But you have to consider that this is the penultimane term in the equation.

[@enginakkent5042]

Besides the random groaning at 00:03 I have something to tell you and I hope you see this. I am an engineering student whose relation with mathematics kind of became sour after entering university ( there are several reasons but I don't want to get into them). Although, I have no idea about how I discovered your videos, I'm very glad that I did. There are something in your videos that made me realise how interesting and enjoying math really is and I should not give up so easily due to my failures or succeses of others. I would like to write more but i don't want it to get more boring. I just want you to know that I owe you a lot. Thank you so much and please keep up with your videos :)

[@mrandersonpw53]

When he started to taking the limit of n, you can asume z fixed (but z except -infinity or infinity) . For zn is a little bit more complicated, but I think it works too using horizontal truncation of lebesgue measure. If z=n, I don't know. He's fine :)

[@yanghwanlim4499]

When deriving the partial integral of the { t^(z-1)(1-t)^n }, it is much simpler to do it in case of n=3 as an example. Then the resulting form clearly hints us about the form in case of n. For the purpose of explaining , I think this way is better. Don't you think?

[@TheLevano22]

9:56 the way you explained it I got the idea that the final product in the brackets would leave us with just the "n+1" term, without it being divided by anything. Every previous term's numerator and following term's denominator will continue to cancel out, including denominator of (n+1)/n term. Leaving us with just "n+1". Someone pls prove me otherwise cause I can't properly continue watching the video without resolving this misunderstanding.

[@AlexHernandez-yj6qe]

Lmao, I'm crossing with the same question. I've started studying the gamma function and saw a similar process but the product is set equal to (n+1) not n. I've tried to make sense out of it, but since I couldn't, I stuck with the other method.

[@9y028]

1:55 You said n is strictly less than z. If you let n go to infinity, does that mean that this only works for z also going to infinity?

[@kevind.shabahang]

Lmao literally the funniest math tutorials! Bravo! Laughed out loud at 7:26 "we are just going to take the limit of this crap"

[@MrRyanroberson1]

7:30 another important theorem to mention is that you have finitely many 1 limits, otherwise you're doing an advanced e.

[@gaetanodagostino7127]

I once read this on a book called "Funzioni Speciali" From L. Gatteschi. Never found that book again.

[@TheNachoesuncapo]

that was a seriously good video!thanks for the outstanding great work as always

[@zactron1997]

8:24 How about the Ganna function?

[@kalvin90210]

Where is the uncut version on your second channel my twink boi???🍄

[@neilgerace355]

9:57 cancelling everything out, you end up with n+1 ...? What did I miss?

[@SvenBeh]

You're right.

[@FlyingOctopus0]

Actually the integral t^(z-1)(1-t)^n is a beta function with parameters z and n+1. beta function can be rewritten in terms of gamma functions as a fraction gamma(z)gamma(n+1)/gamma(z+n+1). I think this relation was used on the channel, but I don't remember in what video.

[@MK-13337]

8:50 "I wanna play more with this junk over here" I bet you do, huh?

[@SidneiMV]

simply awesome definition for the Gamma function!

[@The_Professor_S_]

Was that a 3Blue1Brown-roid?

[@MrRyanroberson1]

nononono 10:05 idk if you realize but you just did n=n+1, which is technically fine for infinitives but is very dubious

[@angelmendez-rivera351]

Ryan Roberson How is it dubious? You literally just said it's fine for infinities. Pick one.

[@tszhanglau5747]

The integral at 17:12 is oddly similar to beta function...I think you could have used it's relationship with gamma function to express the integral in terms of factorials...

[@papsanlysenko5232]

But shouldn't z also approach infinity, as n approaches infty?

[@avinashverma2773]

Yeah, I have the same question

[@Dionisi0]

That's why í disliked this video

[@RieMUisthegoaT]

@@Dionisi0 low brain

[@angelmendez-rivera351]

He said n < z for the sake of simplicity, since it would make breaking the product the way he did easier, but actually, there is no need for n to be strictly smaller.

[@bijoydas6044]

I have a doubt at 9:47 n=(2/1)*(3/2)*(4/3)*..........*(n/n-1)

[@assafabram9649]

Actually it will be wrong to demand that n will be strictly less then z, because you want to take a limit of n to infinity and don't change z.

[@Therealpec96]

Thank you folk your video was very usefull, I hope people agree with me

[@PapaFlammy69]

[@zeroo8756]

I agree

[@Ricocossa1]

"In the real numbers, shit is Abelian." - Flammable Math 2019

[@RieMUisthegoaT]

imo the nth integration by parts was more interesting than the Euler definition of the gamma function lol

[@The_Professor_S_]

Papa’s videos are the best way to start the long ass day at work! And the Gamma function as derived by DADDY EULER? Even better!

[@paulestrada961]

When you go from (n/(n-1)) to (n+1)/n is where I stopped the video. There should be more background as to how you can justify using the fact that n=n can be turned into n=n+1 which of course is not an equality. Saw one of your snack videos and love all of your content. Was curious how someone else did the i! I remembered that you did a video on it because the video I saw from someone else resorted to approximation rather than a closed form and came here by your link on that video.

[@PapaFlammy69]

I made a complementary video on that actually!

[@xHyperElectric]

Where’s the link for the integration by parts uncut version?

[@davidarenas1156]

I have a doubt, when you take the limit as n goes to infinit, shouldn’t you just take it on the last term? Because Its like a progression. Sorry for my bad english.

[@tianvlasic]

That yoke at beginning is very funny😂.

[@rot6015]

Just a comment to increase papa's popularity

[@a.e.6994]

Ganz prima Video!! Danke!!

[@morbidmanatee5550]

Shouldn't the product go from k=1 to n-1 (not n)?

[@asmasaimun3817]

here n

[@agitatethecrystals]

what did you say from 12:27 to 12:29?

[@BigDBrian]

but in the initial series n

[@soldenstoll8495]

Just for fun can you do a video on the integral of sec(x)tan^2(x). It is beautiful because you have to evaluate sec^3(x) which involves coming back to the original integral. Or you could make it a bit harder by doing sqrt(x^2+1).

[@martinmaturanaacevedo4618]

im sorry but, why you can cancel out the z/n as n goes to infinity?, i mean, if n goes to infinity, doesnt that mean that z also goes to infinity?

[@mohibmohib6442]

I suggest you a question please answer gamma (n + 1/2) as a product what is equal to? how to write with the product symbol

[@ronpearson1912]

Why is the Gamma function used for the ao term of the bessel function of the first kind?

[@Fightclub1995]

You need some Hagoromo chalk

[@syedmdabid7191]

Find out the numerical value of Γ(1/3), Γ(1/5) ? Or Γ(1/3) =?, Γ(1/5) =?

[@ajinaajai550]

How can you write a complex number factorial as a real number factorial times's (n+1) .... z ??????

[@bogdancorobean9270]

The Gamma function is cool and everything, but are you ever going to do a video on the Borwein integral? :)

[@bogdancorobean9270]

[@bobus_mogus]

Will you proof "classical" form of gamma function i.e it's integral representation

[@bobus_mogus]

@@PapaFlammy69 thanks :D

[@arnavchaturvedi4818]

For π expansion from '1' to 'n' while expressing 'n' as a finite product you can only pull this upto n-1 as the upper limit then why and how did you go for n as the upper limit. Plz explain.

[@arnavchaturvedi4818]

@@PapaFlammy69 but still, there limit is applied on that basis it's fine to comprehend but here it's just doesn't go through. Help me with this if you can.

[@campbellmorrison8540]

I guess its all about how you look at it

[@goatmatata2798]

Please create a video where u prove gauss's multiplication theorem pleaaaaas

[@OtiumAbscondita]

Shot clock cheese

[@UnordEntertainment]

the limit of ab is not necessarily equal to the limit of a times the limit of b?

[@UnordEntertainment]

also 1/2 * 2/3 * ... * (n+1)/n = (n + 1)/1 = n + 1 which isnt equal to n? EDIT: think it was supposed to be (n+1)^z on the LHS, EDIT 2: gotta finish the vid another time, gotta go to bed

[@ayernee]

how the fuck can you take \(n ightarrow\infty\) if n in an integer less than z??

[@chandranisahanone]

Euler Smells like a GOD🗿🗿🗿🖤🖤🖤

[@trumanburbank6899]

On a related note: If you define, P(x) = integral Γ(x)dx then integrate P(x+1) by parts, you get integral P(x) dx = xP (x) − P (x + 1) All succeeding integrals of P(x) are also algebraic, as are polynomial forms, for example the all integrals of the polynomial sum(m,n)[x^m P^n(x)] can be expressed as a polynomial of a similar form. The same is true if you define Q(x) = integral (1/Γ(x))dx. As you know, 1/Γ(x) is well-behaved. The only thing is, that I don't know what use these functions have.

[@bobthecob8501]

SHOT CLOCK CHEESE!!!

[@amartya-w3i]

Wtf does that mean

[@bobthecob8501]

beta slay My left stroke just went viral

[@bobthecob8501]

Marko Rezic Right stroke put lil' baby in a spiral

[@正大光明-h2p]

我看看这个视频到底能不能让我彻底攻克gamma function

[@duncanw9901]

OOF for the first time I gotta get a pen and paper and try it myself to be convinced it really works :(

[@danielcastillo2299]

You’re building up to something, I know it.

[@meenar5826]

Bro how e to the power -x is 1-x/n to the power n

[@TheFireBrozTFB]

Thats basically the definition of e do the limit of (1+1/n^)^n n--> infinity and youd get e it is very famous and you could see the proof online

[@OtiumAbscondita]

SHOT CLOCK CHEESE

[@robertmunga2630]

Some geometric insight please. Please.

[@mihaipuiu6231]

For me something is wrong in factorial proof! Sorry!

[@tiempoluna1482]

La matemática es independiente del idioma.

[@Павел-и5б3ц]

Класс. Спасибо

[@nikolaalfredi3025]

Last part i.e when you just changed the form of n^z and prdouct of (z+1)(z+2)...(z+n) is not so clear. Also, you must try to be somewhat official....(use good language).

[@正大光明-h2p]

老子还是没懂

[@alissonmelisaruiz6608]

Te amuuuuuuuuuuuuu

[@evanev7]

Spicy

[@kwirny]

You talk german? I mean you kind of sound like one.

[@kwirny]

@@PapaFlammy69 Ich wusste es, auf welcher Uni bist du?

[@kwirny]

@@PapaFlammy69 Ich kann mich nicht entscheiden, Bonn oder Aachen. Will im Wintersemester anfangen.

[@TheAnbyrley]

@@PapaFlammy69 SUNY Potsdam? :-P

[@kwirny]

Ist Bonn geworden :), viel zu tun aufjedenfall.

[@santiagoalvarez622]

too bad

[@GandhiGuru503]

SHOT CLOCK CHEESE!!!

[@OtiumAbscondita]

SHOT CLOCK CHEESE

[@BobBob-ym1vs]

SHOT CLOCK CHEESE!!!

[@markorezic3131]

If I had a nickel for every time I read this today, I would have 5 nickels, just enough to afford the life of people who write that shit

[@OtiumAbscondita]

SHOT CLOCK CHEESE

[@noelabdon4145]

SHOT CLOCK CHEESE!!!

[@markorezic3131]

This is a Monky

[@OtiumAbscondita]

SHOT CLOCK CHEESE

[@OtiumAbscondita]

SHOT CLOCK CHEESE